Error Analysis of Krylov Subspace approximation Based on IDR($s$) Method for Matrix Function Bilinear Forms

The bilinear form of a matrix function, namely $\mathbf{u}^\top f(A) \mathbf{v}$, appears in many scientific computing problems, where $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$, and $f(z)$ is a given analytic function. The Induced Dimension Reduction IDR($s$) method was originally proposed to solve a large-scale linear system, and effectively reduces the complexity and storage requirement by dimension reduction techniques while maintaining the numerical stability of the algorithm. In fact, the IDR($s$) method can generate an interesting Hessenberg decomposition, our study just applies this fact to establish the numerical algorithm and a posteriori error estimate for the bilinear form of a matrix function $\mathbf{u}^{\top} f(A) \mathbf{v}$. Through the error analysis of the IDR($s$) algorithm, the corresponding error expansion is derived, and it is verified that the leading term of the error expansion serves as a reliable posteriori error estimate. Based on this, in this paper, a corresponding stopping criterion is proposed. Numerical examples are reported to support our theoretical findings and show the utility of our proposed method and its stopping criterion over the traditional Arnoldi-based method.